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Information-Theoretic Equilibrium and Observable Thermalization F www.nature.com/scientificreports OPEN Information-theoretic equilibrium and observable thermalization F. Anzà1,* & V. Vedral1,2,3,4,* A crucial point in statistical mechanics is the definition of the notion of thermal equilibrium, which can Received: 13 October 2016 be given as the state that maximises the von Neumann entropy, under the validity of some constraints. Accepted: 31 January 2017 Arguing that such a notion can never be experimentally probed, in this paper we propose a new notion Published: 07 March 2017 of thermal equilibrium, focused on observables rather than on the full state of the quantum system. We characterise such notion of thermal equilibrium for an arbitrary observable via the maximisation of its Shannon entropy and we bring to light the thermal properties that it heralds. The relation with Gibbs ensembles is studied and understood. We apply such a notion of equilibrium to a closed quantum system and show that there is always a class of observables which exhibits thermal equilibrium properties and we give a recipe to explicitly construct them. Eventually, an intimate connection with the Eigenstate Thermalisation Hypothesis is brought to light. To understand under which conditions thermodynamics emerges from the microscopic dynamics is the ultimate goal of statistical mechanics. However, despite the fact that the theory is more than 100 years old, we are still dis- cussing its foundations and its regime of applicability. The ordinary way in which thermal equilibrium properties are obtained, in statistical mechanics, is through a complete characterisation of the thermal form of the state of the system. One way of deriving such form is by using Jaynes principle1–4, which is the constrained maximisation of von Neumann entropy SvN = − Trρ logρ. Jaynes showed that the unique state that maximises SvN (compatibly with the prior information that we have on the system) is our best guess about the state of the system at the equi- librium. The outcomes of such procedure are the so-called Gibbs ensembles. In the following we argue that such a notion of thermal equilibrium, de facto is not experimentally testable because it gives predictions about all possible observables of the system, even the ones which we are not able to measure. To overcome this issue, we propose a weaker notion of thermal equilibrium, specific for a given observable. The issue is particularly relevant for the so-called “Pure states statistical mechanics”5–19, which aims to under- stand how and in which sense thermal equilibrium properties emerge in a closed quantum system, under the assumption that the dynamic is unitary. In the last fifteen years we witnessed a revival of interest in these ques- tions, mainly due to remarkable progresses in the experimental investigation of isolated quantum systems20–25. The high degree of manipulability and isolation from the environment that we are able to reach nowadays makes possible to experimentally investigate such questions and to probe the theoretical predictions. The starting point of Jaynes’ derivation of statistical mechanics is that SvN is a way of estimating the uncertainty that we have about which pure state the system inhabits. Unfortunately we know from quantum information theory that it does not address all kind of ignorance we have about the system. Indeed, it is not the entropy of an observable (though the state is observable); its conceptual meaning is not tied to something that we can measure. This issue is intimately related with the way we acquire information about a system, i.e. via measurements. The process of measuring an observable on a quantum system allows to probe only the diagonal part of the density matrix λρiiλ , when this is written in the observable eigenbasis {}λi . For such a reason, from the experimental point of view, it is not possible to assess whether a many-body quantum system is at thermal equilibrium (e.g. Gibbs state ρG): the number of observables needed to probe all the density matrix elements is too big. In any experimentally reasonable situation we have access only to a few (sometimes just one or two) observables. It is 1Atomic and Laser Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK. 2Centre for Quantum Technologies, National University of Singapore, 117543, Singapore. 3Department of Physics, National University of Singapore, 2 Science Drive 3, 117551, Singapore. 4Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, 100084, Beijing, China. *These authors contributed equally to this work. Correspondence and requests for materials should be addressed to F.A. (email: fabio.anza@ physics.ox.ac.uk) SCIENTIFIC REPORTS | 7:44066 | DOI: 10.1038/srep44066 1 www.nature.com/scientificreports/ therefore natural to imagine situations in which the outcomes of measurements are compatible with the assump- tion of thermal equilibrium, while the rest of the density matrix of the system is not. Despite that, we think that the fact that a distribution is compatible with its thermal counterpart will lead to the emergence of certain thermal properties, concerning the specific observable under scrutiny. Building our intuition on that, we propose a new notion of thermal equilibrium specific for a given observable, experimentally verifiable and which relies on a figure of merit that is not the von Neumann entropy. A good choice for such a figure of merit comes from quantum information theory and it is the Shannon entropyH of the eigenvalues 26 probability distribution {p(λj)} of an observable . The well-known operational interpretation ofH matches our needs since it addresses the issue of the knowledge of an observable and it provides a measure for the entropy of its probability distribution. Throughout the paper we will work under the assumption that the Hilbert space of the system has finite dimension and we will refer to the case in which the Hamiltonian of the system has no local conserved quantities, even though it is possible to address situations where there are several conserved quantities, like integrable quan- tum systems. We will also assume that the observable has a pure-point spectrum with the following spectral decomposition =∑j λ jjΠ , where Πj is the projector onto the eigenspace defined by the eigenvalue λj. H is the entropy of its eigenvalues probability distribution p(λj) ≡ Tr(ρΠj ) Hp[]ρλˆ ≡− ∑ ()jjlog(p λ ) ∈ j:λσj (1) where σ is the spectrum of . We propose to define the notion of thermal equilibrium, for an arbitrary but fixed observable , via a charac- terisation of the probability distribution of its eigenvalues. We will say that is at thermal equilibrium when its eigenvalues probability distribution p(λj) maximises the Shannon entropy H, under arbitrary perturbations with conserved energy. We call an observable with such a probability distribu- tion: thermal observable. It is important to note that this notion characterises only the probability distribution at equilibrium and it does eq not uniquely identify an equilibrium state. Given the equilibrium distribution p(λj) = p (λj), there will be several quantum states ρ which give the same probability distribution for the eigenvalues λj. In this sense this is a weaker notion of equilibrium, with respect to the ordinary one. In the rest of the paper we study the main consequences of the proposed notion of observable-thermal-equilbrium: its physical meaning and the relation with Gibbs ensembles. The investigation will show that the proposed notion of equilibrium is able to address the emergence of thermalisation. This is our first result. Furthermore, we study the proposed notion of equilibrium in a closed quantum system and prove that there is a large class of bases of the Hilbert space which always exhibit thermal behaviour and we give an algorithm to explicitly construct them. We dub them Hamiltonian Unbiased Bases (HUBs) and, accordingly, we call an observable which is diagonal in one of these bases Hamiltonian Unbiased Observable (HUO). The existence and precise characterisation of observables which always thermalise in a closed quantum system is our second result. Furthermore, we investigate the relation between the notion of thermal observable and one of the main par- adigms of pure states statistical mechanics: the Eigenstate Thermalisation Hypothesis (ETH)27–37. We find an intimate connection between the concept of HUOs and ETH: the reason why these observables thermalise is pre- cisely because they satisfy the ETH. Hence, with the existence and characterisation of the HUOs we are providing a genuine new prediction about which observables satisfies ETH, for any given Hamiltonian. The existence of this relation between HUOs and ETH is a highly non trivial feature and the fact that we can use it to predict which observables will satisfy ETH is our third result. In the conclusive section we summarise the results and discuss their relevance for some open questions. Results Information-theoretic equilibrium. The request that the equilibrium distribution must be a maximum for H is phrased as a constrained optimisation problem and it is solved using the Lagrange multipliers tech- nique. The details are given in in the Methods section. Two sets of equilibrium equations are obtained and we now show how they account for the emergence of thermodynamic behaviour in the observable . We assume that the only knowledge that we have on the system is the normalisation of the state and the mean value of the energy 〈 T〉 = E0, where T is the Hamiltonian of the system. The Hamiltonian has the following spec- tral decomposition T = ∑ αEαTα, where T αα≡ EEα and we assume that its eigenvectors {}Eα provide a full basis of the Hilbert space. We call ψn the eigenstates of the density operator, ρn are the respective projectors and qn its eigenvalues. To describe the state of the system we use the following convenient basis: {,js} in which the first index j runs over different eigenvaluesλ j of and the second index s accounts for the fact that there might be degeneracies.
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